Related papers: Hyperbolic Gradient Flow: Evolution of Graphs in R…
We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in…
In this paper, we introduce a geometric flow for Lagrangian submanifolds in a K\"ahler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary…
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is…
We consider the evolution of a $n$-dimensional convex hypersurface in the euclidean space under mean curvature flow with densities $e^{\varepsilon \frac12 n\mu^2 |x|^2}$, $\varepsilon =\pm 1$, and completely determine it depending on the…
During the past thirty years hyperbolic type metrics have become popular tools also in modern mapping theory, e.g., in the study of quasiconformal and quasiregular maps in the euclidean $n$-space. We study here several metrics that one way…
Generating graphs with hierarchical structures remains a fundamental challenge due to the limitations of Euclidean geometry in capturing exponential complexity. Here we introduce \textbf{GGBall}, a novel hyperbolic framework for graph…
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic $3$ manifolds are…
Scene graph representations enable structured visual understanding by modeling objects and their relationships, and have been widely used for multiview and 3D scene reasoning. Existing methods such as MSG learn scene graph embeddings in…
We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov's method, we derive an L1 error estimate which applies to a…
While Hyperbolic Graph Neural Network (HGNN) has recently emerged as a powerful tool dealing with hierarchical graph data, the limitations of scalability and efficiency hinder itself from generalizing to deep models. In this paper, by…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
This paper concerns the validity of the Prandtl boundary layer theory for steady, incompressible Navier-Stokes flows over a rotating disk. We prove that the Navier Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid…
In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean $n$-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support…
We study integral transforms mapping a function on the Euclidean space to the family of its integration on some hypersurfaces, that is, a function of hypersurfaces. The hypersurfaces are given by the graphs of functions with fixed axes of…
Flows of vector fields are an essential tool in differential geometry, with countless applications in both theory and practice. While they have been extensively studied for ordinary manifolds and supermanifolds, a treatment of flows in…
Geometric representation learning has recently shown great promise in several machine learning settings, ranging from relational learning to language processing and generative models. In this work, we consider the problem of performing…
We study contracting curvature flows of compact hypersurfaces with positive sectional curvature in hyperbolic space $\mathbb{H}^{n+1}$. The speed is assumed to be homogeneous of degree one in the principal curvatures and to satisfy certain…
For a mean curvature flow of complete graphical hypersurfaces $M_{t}=\operatorname{graph} u(\cdot,t)$ defined over domains $\Omega_{t}$, the enveloping cylinder is $\partial\Omega_{t}\times\mathbb{R}$. We prove the smooth convergence of…
In this paper, we consider the high order geometric flows of a submanifolds $M$ in a complete Riemannian manifold $N$ with $\dim(N)=\dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and…
In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and…